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Triangle Proof

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Triangle Proof by Kathy McDonald section 3.1 #7 Triangle Proof by Kathy McDonald section 3.1 #7 By assumption, the original triangle has n segments on each side And ... – PowerPoint PPT presentation

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Title: Triangle Proof


1
Triangle Proof
  • by Kathy McDonald
  • section 3.1 7

2
Prove When dividing each side of an equilateral
triangle
3
into n segments
4
then connecting the division points with all
possible segments parallel
to the original sides, n² small triangles are
created.
5
Proof by induction
Let S n?N f(n) n²
6
Show 1 ?S
1
f(n) n² f(1) 1 1²
7
Show 2 ?S
when dividing each side into 2 segments
8
and connecting division points as described,
9
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10
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11
4 small triangles are created.
12
f(n) n² f(2) 4 2²
13
Show 3 ?S
when dividing each side into 3 segments
14
and connecting division points as described,
15
(No Transcript)
16
(No Transcript)
17
9 small triangles are created.
18
f(n) n² f(3) 9 3²
19
Assume n ?S.
Assume when dividing each side into n segments
and connecting division points as described, n²
small triangles are created. Assume f(n) n².
20
Show n1 ?S.
Show when dividing each side into n1 segments
and connecting division points as described,
(n1)² small triangles are created. Show f(n1)
(n1)².
21
Consider a divided triangle
with n segments on each side.
22
When a segment equal in size to the n segments is
added to each side
23
and those endpoints are connected,
24
a space is created at the bottom of the original
triangle.
Also, a new, bigger equilateral triangle has
been created.
25
This new, bigger triangle has n1 segments on
each side.
n segments

1 segment
26
Now, the parallel dividing lines are extended
down
to the base of the new, bigger triangle.
27
More small triangles are created.
28
The n segments of the base of the original
triangle
29
correspond to n bases of the new, small triangles
created.
30
Also, the n1 segments of the base of the new,
bigger triangle
31
correspond to n1 bases of the new, small
triangles.
32
So, n(n1) bases
33
correspond to n(n1) new, small triangles
34
By assumption, the original triangle has n
segments on each side
And n² small triangles inside.
35
By adding 1 segment to each side of this triangle,
n (n1) small triangles are added.
36
The total small triangles of the new, bigger
triangle is
37
n² n (n1)
n²2n1 (n1)(n1)
(n1)²
38
This shows n1 ?S.
By induction, S ? N.
39
Dwight says, thats it.
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